{
  "schema": "AN04-restored-simultaneous-reflections-proof-map/v1",
  "proofs": [
    {
      "id": "SI:F1",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "## 2.",
      "dependencies": [
        "FD:R2",
        "SN:F0",
        "G2:NF1",
        "G2:NF7",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:P15.1",
        "U001:P3"
      ],
      "scope": "First normal form from explicit transverse-flow integrals, invertible coordinate differential and weighted matching"
    },
    {
      "id": "SI:W1",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "**Weighted-remainder verification.**",
      "dependencies": [
        "FD:D1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:F0-DIFF"
      ],
      "scope": "Preservation of unequal smooth remainder divisibility under both compositions and parity-preserving inverse conjugacy"
    },
    {
      "id": "SI:J1",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "**Lemma 3.1",
      "dependencies": [
        "SI:F1",
        "SI:W1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Even weighted coefficients vanish; odd errors eliminated with exact two homological equations and inverse remainder order"
    },
    {
      "id": "SI:J2",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "**Finite-jet justification.**",
      "dependencies": [
        "SI:J1",
        "SI:W1",
        "U001:P3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Compatible stabilization of every finite normal jet, its composition/inverse and common tangential domain"
    },
    {
      "id": "SI:J3",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "**Lemma 3.2",
      "dependencies": [
        "U001:P14.3",
        "U001:P13.2-series",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full local parameter Borel realization, parity, all derivative bounds, FTC termwise differentiation and exact jets"
    },
    {
      "id": "SI:J4",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "Apply the lemma to every coordinate component",
      "dependencies": [
        "SI:J2",
        "SI:J3",
        "U001:P3"
      ],
      "scope": "Actual parity-preserving diffeomorphism and equality of all normal and mixed jets after conjugation"
    },
    {
      "id": "SI:I0",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "**Lemma 4.1",
      "dependencies": [
        "SI:J4",
        "U001:P14.3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Compact extension preserving flat jets and uniform arbitrary-power bounds in all derivatives"
    },
    {
      "id": "SI:I1",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "We first control a number of iterates",
      "dependencies": [
        "SI:I0",
        "U001:F0-DIFF"
      ],
      "scope": "Exact summed nilpotent-shear recurrence and simultaneous normal/derivative bootstrap for O(1/abs(s)) iterates"
    },
    {
      "id": "SI:I2",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "For initial \\(|t|\\le1\\)",
      "dependencies": [
        "SI:I1"
      ],
      "scope": "Uniform finite escape, monotone sign, no return and locally uniform finite stopping time off the fixed hypersurface"
    },
    {
      "id": "SI:I3",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "Define, for \\(s\\ne0\\)",
      "dependencies": [
        "SI:I2",
        "SI:I1",
        "SI:I0",
        "U001:F0-DIFF"
      ],
      "scope": "Locally finite correction series, all mixed flat derivatives, smooth boundary extension and exact telescoping on a common germ domain"
    },
    {
      "id": "SI:C1",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "## 5.",
      "dependencies": [
        "FD:R2"
      ],
      "scope": "Invariant-function identity and exact parity averages for both involutions"
    },
    {
      "id": "SI:C2",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "For each spectator coordinate",
      "dependencies": [
        "SI:C1",
        "SI:I3",
        "SI:J4",
        "FD:D1",
        "U001:P3",
        "U001:F0-DIFF"
      ],
      "scope": "Actual spectator/normal correction, preservation of flatness under composition and inverse, and exact joint parity"
    },
    {
      "id": "SI:C3",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "The remaining defect",
      "dependencies": [
        "SI:C2",
        "SI:I3",
        "U001:P3"
      ],
      "scope": "Final tangential flat correction with both ordered signs, even average and actual simultaneous coordinate theorem"
    },
    {
      "id": "SI:M0",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "**Theorem 1.1",
      "dependencies": [
        "SI:C3",
        "FD:R2",
        "U001:P9.1"
      ],
      "scope": "Full ordinary simultaneous theorem, exact reflection eigenlines, intrinsic tangential line and opposite ordered shear products"
    },
    {
      "id": "SI:S1",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "**Corollary 6.1",
      "dependencies": [
        "SI:M0",
        "U001:P9.1"
      ],
      "scope": "Invariant transverse slice retaining both reflection lines and their common fixed hypersurface"
    },
    {
      "id": "SI:H1",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "**Theorem 6.2",
      "dependencies": [
        "SI:S1",
        "SI:M0",
        "U001:P3",
        "U001:L5.4.1",
        "U001:F0-DIFF"
      ],
      "scope": "Full homogeneous theorem under the equivariant-cone definition: unique saturated slice product, exact radial coordinate and both homogeneous conjugacies"
    },
    {
      "id": "SI:X1",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "**Exercise 7.1 ",
      "dependencies": [
        "SI:M0"
      ],
      "scope": "Original Exercise 7.1 and complete unchanged solution"
    },
    {
      "id": "SI:X2",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "**Exercise 7.2 ",
      "dependencies": [
        "SI:F1",
        "U001:L5.4.1",
        "U001:P15.1"
      ],
      "scope": "Original Exercise 7.2 and complete unchanged solution"
    },
    {
      "id": "SI:X3",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "**Exercise 7.3 ",
      "dependencies": [
        "SI:J1",
        "SI:M0",
        "U001:P3"
      ],
      "scope": "Original Exercise 7.3 and complete unchanged solution"
    },
    {
      "id": "SI:X4",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "**Exercise 7.4 ",
      "dependencies": [
        "SI:J1"
      ],
      "scope": "Original Exercise 7.4 and complete unchanged solution"
    },
    {
      "id": "SI:X5",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "**Exercise 7.5 ",
      "dependencies": [
        "SI:J1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Original Exercise 7.5 and complete unchanged solution"
    },
    {
      "id": "SI:X6",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "**Exercise 7.6 ",
      "dependencies": [
        "SI:C3",
        "U001:P14.3"
      ],
      "scope": "Original Exercise 7.6 and complete unchanged solution"
    },
    {
      "id": "SI:X7",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "**Exercise 7.7 ",
      "dependencies": [
        "SI:I3",
        "U001:P14.3"
      ],
      "scope": "Original Exercise 7.7 and complete unchanged solution"
    },
    {
      "id": "SI:X8",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "**Exercise 7.8 ",
      "dependencies": [
        "SI:S1"
      ],
      "scope": "Original Exercise 7.8 and complete unchanged solution"
    },
    {
      "id": "SI:X9",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "466d0830644c10874905543b6ac6a75e7de4f43880f6d415352f55d4af8a8fb5",
      "proof_locator": "**Exercise 7.9 ",
      "dependencies": [
        "SI:H1",
        "SI:M0"
      ],
      "scope": "Original Exercise 7.9 and complete unchanged solution"
    }
  ],
  "earlier_maps": [
    "../20261004-free-stationary-phase/proof-map.json",
    "../20261004-free-intrinsic-graph/proof-map.json",
    "../20261004-free-tangent-zoom/proof-map.json",
    "../20261004-free-canonical-composition/proof-map.json",
    "../20261005-restored-oscillatory/proof-map.json",
    "../20261005-restored-phase-space/proof-map.json",
    "../20261005-restored-phase-equivalence/proof-map.json",
    "../20261005-restored-intrinsic-regularity/proof-map.json",
    "../20261005-restored-tangent-zoom/proof-map.json",
    "../20261005-restored-gaussian-symbols/proof-map.json",
    "../20261005-restored-linear-reduction/proof-map.json",
    "../20261005-restored-analytic-composition/proof-map.json",
    "../20261005-restored-subprincipal-transport/proof-map.json",
    "../20261005-restored-graph-egorov/proof-map.json",
    "../20261005-restored-corank-continuity/proof-map.json",
    "../20261005-restored-submanifolds/proof-map.json",
    "../20261005-restored-cubic-scaling/proof-map.json",
    "../20261005-restored-smooth-descent/proof-map.json"
  ],
  "transitive_U001_ids": [
    "A0",
    "CLOSED-BALL-CONTRACTION",
    "D0",
    "E0",
    "EXPONENTIAL-TRIGONOMETRIC",
    "F0-CALC",
    "F0-COMP",
    "F0-DIFF",
    "FTC-TAYLOR-COMPACT-PARAMETERS",
    "I0",
    "L11.4.2",
    "L2.1.10-inc",
    "L2.1.17",
    "L2.2.1",
    "L2.2.11",
    "L2.2.3",
    "L2.2.5",
    "L2.3.2",
    "L2.3.4-upper",
    "L2.3.5",
    "L2.3.8",
    "L2.4.5",
    "L2.6.5",
    "L3.1.7",
    "L3.3.7",
    "L3.3.8",
    "L4.2.2",
    "L4.2.3",
    "L4.2.4",
    "L4.2.6",
    "L5.1.10",
    "L5.1.11",
    "L5.1.13",
    "L5.1.2",
    "L5.1.7",
    "L5.1.8",
    "L5.2.1",
    "L5.2.2",
    "L5.2.4-positive",
    "L5.2.6",
    "L5.2.7",
    "L5.3.1",
    "L5.3.3",
    "L5.3.5",
    "L5.4.1",
    "L5.4.2",
    "L7.1.4",
    "L7.1.5",
    "L7.2.19",
    "L7.2.22",
    "L7.2.6",
    "L7.2.9-open",
    "L7.3.11",
    "L7.3.12",
    "L7.3.9",
    "L7.4.10",
    "L7.4.11",
    "L7.4.2",
    "L7.4.4",
    "L7.4.9",
    "L7.5.11",
    "L7.5.12",
    "L7.5.2",
    "L7.5.5",
    "L7.5.6",
    "L7.6.2",
    "L8.1.13",
    "L8.1.14",
    "L8.1.16",
    "L8.1.17",
    "L8.1.18",
    "L8.2.4",
    "L8.2.5",
    "L8.2.6",
    "L8.2.7",
    "L8.2.8",
    "L8.2.9",
    "L8.3.2",
    "L8.3.3",
    "L8.3.5",
    "L8.3.6",
    "L8.3.7",
    "L8.3.9",
    "L8.4.1",
    "L8.4.2",
    "L8.4.6",
    "L8.5.1",
    "L8.5.6",
    "L8.6.2",
    "L9.1.1",
    "P1",
    "P10.1",
    "P10.2",
    "P10.2-typo",
    "P10.3",
    "P10.4",
    "P10.5-rules",
    "P10.6",
    "P10.6-zero",
    "P11.1",
    "P11.2-bridge",
    "P11.2-orders",
    "P12.1",
    "P12.2-criterion",
    "P12.2-upper",
    "P12.3",
    "P12.4",
    "P12.5-endpoints",
    "P12.5-extension",
    "P12.6",
    "P12.7",
    "P13.1",
    "P13.2-mertens",
    "P13.2-series",
    "P14.2-inverse",
    "P14.2-powers",
    "P14.3",
    "P15.1",
    "P15.2",
    "P16.1",
    "P16.2",
    "P16.3",
    "P2",
    "P3",
    "P6.0",
    "P6.1",
    "P6.2",
    "P6.3-inf",
    "P6.3-subtract",
    "P6.3-tail",
    "P6.4-ball",
    "P6.4-complete",
    "P6.5",
    "P6.6",
    "P7",
    "P8-root",
    "P8-smooth",
    "P9.1",
    "P9.2",
    "P9.2-operator",
    "P9.3"
  ],
  "transitive_earlier_package_ids": [
    "FD:D1",
    "FD:R2",
    "G2:NF0",
    "G2:NF1",
    "G2:NF2",
    "G2:NF3",
    "G2:NF4",
    "G2:NF5",
    "G2:NF6",
    "G2:NF7",
    "SN:F0"
  ],
  "bounded_lesson_P514_closed": true,
  "human_review_complete": false,
  "full_course_complete": false,
  "public_release_authorized": false
}